From bi-ideals to periodicity

DOI:10.1051/ita:2008010 www.rairo-ita.org

FROM BI-IDEALS TO PERIODICITY

J¯ anis Buls

and Aivars Lorencs

Abstract. The necessary and sufficient conditions are extracted for periodicity of bi-ideals. They cover infinitely and finitely generated bi-ideals.

Mathematics Subject Classification.68R15, 94A55, 68Q15.

1. Introduction

The periodicities are fundamental objects, due to their primary importance in word combinatorics [8,9] as well as in various applications. The study of periodicities is motivated by the needs of molecular biology [6] and computer science. Particularly, we mention here such fields as string matching algorithms [4], text compression [13] and cryptography [11].

In different areas of mathematics, people consider a lot of hierarchies which are typically used to classify some objects according to their complexity. Here we deal with the hierarchy

B⊃P, where Bis the class of bi-ideals,

Pis the class of periodic words.

This hierarchy comes from combinatorics on words, where these classes are being investigated intensively (cf. [2,8–10]). Bi-ideal sequences have been considered, with different names, by several authors in algebra and combinatorics [1,3,7,12,14].

Every bi-idealx is the limit of some bi-ideal sequence (vi). This bi-ideal se- quence can be represented uniquely by the sequence (ui), where v0 = u0 and

Keywords and phrases.Periodic words, bi-ideals, the sequence generates the bi-ideal, finitely generated bi-ideals.

1 Department of Mathematics, University of Latvia, Rai¸na bulv¯aris 19, R¯ıga, 1586, Latvia;

[email protected]; web site: http://home.lanet.lv/~buls

2 Institute of Electronics and Computer Science, Dz¯erbenes street 14, R¯ıga, 1006, Latvia;

[email protected]

Article published by EDP Sciences c EDP Sciences 2008

∀i≥0vi+1=viui+1vi. We characterize the periodic words through this represen- tation. At first we give an exhaustive description (Th. 3.7) of periodicity for all classes of bi-ideals. Then for periodic bi-ideals we demonstrate if everyui appears infinitely often then every ui is a power of the certain word. This leads to the effective method for finitely generated bi-ideals to check whether the bi-ideals are periodic.

2. Preliminaries

In this section we present most of the notations and terminology used in this paper. Our terminology is more or less standard (cf. [10]) so that a specialist reader may wish to consult this section only if need arise.

LetAbe a finite non-empty set andA^∗ the free monoid generated byA. The set Ais also called analphabet, its elementsletters and those of A^∗ finite words.

The role of the identity element is performed by theempty wordwhich is denoted byλ. We setA⁺=A^∗\{λ}.

A word w ∈ A⁺ can be written uniquely as a sequence of letters as w = w1w2. . . wl, with wi∈A, 1≤i≤l, l >0. The integerl is called the lengthofw and denoted|w|. The length of λis 0. We setw⁰=λand∀i wⁱ⁺¹=wⁱw;

w⁺= ∞ i=1

{wⁱ}, w^∗ =w⁺∪ {λ}.

A positive integerpis called aperiodofw=w1w2. . . wlif the following condition is satisfied:

1≤i≤l−p ⇒ wi=wi+p.

We recall the important periodicity theorem due to Fine and Wilf [5]:

Theorem 2.1. Let w be a word having periods pand q and denote by gcd(p, q) the greatest common divisor ofpandq. If|w| ≥p+q−gcd(p, q), thenwhas also the periodgcd(p, q).

The word w ∈ A^∗ is called a factor (or subword) of w ∈ A^∗ if there exist u, v ∈ A^∗ such that w = uwv. The word u (respectively v) is called a prefix (respectively asuffix) ofw. The ordered triple (u, w, v) is called anoccurrenceof w inw. The factorw is called aproperfactor ifw=w. We denote respectively byF(w), Pref(w) and Suff(w) the sets ofwfactors, prefixes and suffixes.

An (indexed) infinite wordxon the alphabet A is any total mapx : N→A. We set for anyi≥0,xi=x(i) and write

x= (xi) =x0x1. . . xn. . . The set of all the infinite words overAis denoted byA^ω.

The word w ∈ A^∗ is a factor of x∈ A^ω if there exist u∈ A^∗, y ∈ A^ω such thatx=uwy. The wordu(respectivelyy) is called aprefix(respectively asuffix)

of x. We denote respectively byF(x), Pref(x) and Suff(x) the sets ofxfactors, prefixes and suffixes. For any 0≤m≤n, bothx[m, n] andx[m, n+ 1) denote a factorxmxm+1. . . xn. The indexed wordx[m, n] is called anoccurrenceofw inx ifw =x[m, n]. The suffixxnxn+1. . . xn+i. . .is denoted byx[n,∞).

Ifv ∈A⁺ we denote byv^ω the infinite wordv^ω =vv . . . v . . .This wordv^ω is called aperiodic word. The concatenation ofu=u1u2. . . uk ∈A^∗ andx∈A^ω is the infinite word

ux=u1u2. . . ukx0x1. . . xn. . .

A word xis called ultimately periodic if there exist words u∈A^∗, v ∈A⁺ such thatx=uv^ω. In this case,|u|and|v|are called, respectively, an anti-periodand aperiodofx.

A sequence of words ofA^∗

v0, v1, . . . , vn, . . .

is called a bi-ideal sequence if ∀i ≥ 0 (vi+1 ∈ viA^∗vi). The term “a bi-ideal sequence” is due to the fact that ∀i≥0 (viA^∗vi) is a bi-ideal ofA^∗.

Corollary 2.2. Let (vn)be a bi-ideal sequence. Then vm∈Pref(vn)∩Suff(vn) for allm≤n.

A bi-ideal sequencev0, v1, . . . , vn, . . .is calledproperifv0 =λ. In the following the term bi-ideal sequence will be referred only to proper bi-ideal sequences.

Ifv0, v1, . . . , vn, . . .is a bi-ideal sequence, then there exists a unique sequence of words

u0, u1, . . . , un, . . . such that

v0=u0, ∀i≥0 (vi+1=viui+1vi).

Let us consideru, v∈A^∞=A^∗∪A^ω. Thend(u, v) = 0 ifu=v, otherwise d(u, v) = 2⁻ⁿ,

where

n= max{ |w| |w∈Pref(u)∩Pref(v)}. It is called aprefix metric.

Let v0, v1, . . . , vn. . . be an infinite bi-ideal sequence, where v0 = u0 and

∀i ≥ 0 (vi+1 = viui+1vi). Since for all i ≥ 0 the word vi is a prefix of the next wordvi+1 the sequence (vi) converges, with respect to the prefix metric, to the infinite wordx∈A^ω

x=v0(u1v0)(u2v1). . .(unvn−1). . .

This wordxis called abi-ideal. We say the sequence (ui)generatesthe bi-idealx.

Convention. Let x be a bi-ideal generated by (ui), then x = lim

i→∞vi, where v0=u0 andvi+1=viui+1vi. We adopt this notational convention henceforth.

Letxbe an infinite word. A factoruofxis calledrecurrentif it occurs infinitely often inx. The wordxis calledrecurrentwhen any of its factors is recurrent.

Proposition 2.3. (see, e.g., [10]) A word x is recurrent if and only if it is a bi-ideal.

Lemma 2.4. (see, e.g., [10]) Let x∈A^ω be an ultimately periodic word. Ifxis recurrent, then xis periodic.

Due to this lemma we can restrict ourselves. Therefore we investigate only the periodicity of bi-ideals and say nothing about ultimate periodicity.

3. The periodicity of bi-ideals

The following three lemmas are very easy, but they turn out to be extremely useful:

Lemma 3.1. If x=w^ω and T is the minimal period of the word x, then T\|w|, i.e. T divides |w|.

Proof. Letn=T|w|, then bothT and|w|are periods of the wordx[0, n). Hence (Th.2.1)t= gcd(T,|w|) is a period ofx[0, n). Now we have

∀i x[0, n) =x[ni, n(i+ 1)).

Thereforet is a period ofx. Since T is the minimal period of the word x, then t≥T ≥gcd(T,|w|) =t. HenceT = gcd(T,|w|), therebyT\|w|.

Lemma 3.2. If x=w^ω=uvyand|w|=|v|, thenvy=y=v^ω.

Proof. Let|w| =t and |u| =k+ 1, then v =xk+1xk+2. . . xk+t, since |v| =|w|.

We have∀i xi+t=xi, therefore

∀j∈1, t∀s x_k+j=xk+j+st.

Lemma 3.3. If ∃u∈ A⁺ ux = x∈ A^ω, then a word x is periodic with the minimal periodT\|u|.

Proof. Letu=a1a2. . . at−1, where∀j aj∈A, and y=ux, then

∀i xi=yi+t. Let

y=ux=x . Hence

∀i y_i =xi=yi+t.

This means thatyis periodic with a periodt. Sincey=x, thenxis periodic with a period t too. Let T is the minimal period of x, then by Lemma 3.1 T\t, i.e.

T\|u|.

Corollary 3.4. Let |v| be the minimal period ofx=v^ω. If v=x[k, k+|v|) then |v|\k.

Proof. If, for any k,v=x[k, k+|v|), then (see Lem.3.2) x=x[0, k)v^ω=x[0, k)x.

Hence by Lemma3.3|v|\|x[0, k)|=k.

Lemma 3.5. If existsn such thatvnu∈v^∗ and∀i∈Z₊(un+i∈uv^∗), then

∀i∈N(vn+i∈v^∗vn). Proof. Ifi= 0 thenvn+i=vn=λvn∈v^∗vn.

Further, we shall prove the lemma by induction on i, i.e., suppose that vn+i ∈v^∗vn, namely,

∃k∈N(vn+i=v^kvn). By assumption,vnu∈v^∗ andun+i+1∈uv^∗,i.e.

∃l∈N(vnu=v^l) ∧ ∃m∈N(un+i+1=uv^m). Hence

vn+i+1 = vn+iun+i+1vn+i= (v^kvn)(uv^m)(v^kvn)

= v^k(vnu)v^m+kvn=v^kv^lv^m+kvn∈v^∗vn.

We have completed the inductive step.

Lemma 3.6. If tis the period of the bi-ideal xand|vn| ≥t, then

∀i∈Z₊un+1x=un+ix .

Proof. We havevn+i=vn+i−1un+ivn+i−1. Hence, ifi∈Z+ then (Cor.2.2)

∀i∈Z+∃v_i vn+i =vnv_ivn. Now, by definition ofx

x = vnun+1vn. . .

x = vn+iun+i+1vn+i. . .=vnv_ivnun+i+1vn. . . By assumption,xis periodic, therefore

x=v^ω, where |v|=t .

Sincev∈Pref(vn) then by Lemma3.2

x = vnun+1x , x = vnun+i+1x .

Hence∀i∈Z+x=vnun+ix .Thus∀i∈Z+ un+1x=un+ix . Theorem 3.7. A bi-idealxis periodic if and only if

∃n∈N∃u∃v(vnu∈v^∗ ∧ ∀i∈Z+un+i∈uv^∗).

Proof. ⇒LetT be the minimal period of the wordx, then∃n∈N|vn| ≥T. Thus by Lemma3.6

∀i∈Z+ un+1x=un+ix . Letube the longest word of the set_∞

i=1Pref(un+i) then

∀i∈Z+∃u_i(un+i=uu_i). Particularly,∃k un+k=u. This means that

∀i∈Z₊ uu_ix=un+ix=un+kx=ux . Thus

∀i∈Z₊ u_ix=x . Hence by Lemma3.3

∀i∈Z₊ T\|u_i|. Thereby

∀i∈Z+ u_i∈v^∗, wherev=x[0, T). Thus

∀i∈Z+ un+i=uu_i∈uv^∗. Note

x=vnun+1vn. . .=vnuu₁vn. . .

Sinceu₁ ∈v^∗ and v ∈Pref(vn), then [Lemma3.2] x=vnux. Hence [Lem.3.3]

vnu∈v^∗.

⇐By Lemma3.5

∀i∈N∃ki∈Nvn+i=v^kivn. Since lim

k→∞|vk|=∞then lim

i→∞ki=∞. Thus x= lim

k→∞vk = lim

i→∞vn+i= lim

i→∞v^kivn =v^ω.

4. Powers

Observation. If allui∈w^∗ for some wordw =λ, then the bi-ideal generated by (ui) is periodic.

The following example demonstrates the converse is not true in general.

Example 4.1. Letxbe the bi-ideal generated by (ui), where u0 = 0,

u1 = 1,

∀i >1 ui = 00100. Then

v0 = 0, v1 = 010,

v2 = 010 00100 010,

v3 = 01000100010 00100 01000100010, . . .

andx= lim

i→∞vi= (0100)^ω. Thusxis periodic.

Nevertheless, if every uj appears infinitely often in (ui), then the converse is valid.

Theorem 4.2. Let(ui)be a sequence of words, which contains everyuj infinitely often. The bi-idealxgenerated by (ui)is periodic if and only if

∃w∀i ui∈w^∗.

Proof. ⇒Letxbe a periodic bi-ideal, then by Theorem3.7

∃n∈N∃u∃v(vnu∈v^∗ ∧ ∀i∈Z+un+i∈uv^∗).

Hence by Lemma3.5 |v|is the period ofx. Therefore we can assume that |v|is the minimal period ofxand|u|<|v|. Since the sequence (ui) contains everyuj

infinitely often then by Theorem3.7 ∀i∈N(ui ∈uv^∗).

Now suppose thatui=ufor alli < mbutum=uv^k, wherek >0. Then there existα∈Z+ andy such that

x=u^αv^ky . (i) Ifu=λthen∀i ui∈v^∗.

(ii) Otherwiseu=λ. Then (Corollary3.4)|v|\α|u|. Hence, there existsβ∈Z+

such thatα|u|=β|v|. Thusx=v^ω=u^ω. Contradiction, since|u|<|v|and|v|is the minimal period ofx.

⇐See Observation.

Now we turn our attention to the problem of effectiveness.

Definition 4.3. Assume that (ui) generates a bi-idealx. The bi-idealxis called finitely generatedif

∃m∀i∀j(i≡j(modm)⇒ui =uj).

In this situation, we say that the m–tuple (u0, u1, . . . , um−1) generates the bi- idealx.

Theorem 4.4. A bi-ideal x generated by (u0, u1, . . . , um−1) is periodic if and only if

∃w∀i∈0, m−1 ui∈w^∗.

Proof. As a corollary from Definition4.3and Theorem4.2.

This theorem gives a method to generate nonperiodic bi-ideals. Let (u0, u1, . . . , um−1)

be anym-tuple chosen at random. Letv be any shortest word from the set {u0, u1, . . . , um−1}

andw be the shortest prefix of v such thatv ∈w⁺. If there exists ui such that ui∈/ w^∗then the bi-ideal generated by (u0, u1, . . . , um−1) is not periodic. This can be easily checked by a deterministic algorithm.

Acknowledgements. The useful suggestions of two referees are gratefully acknowledged.

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